Difference between revisions of "1975 AHSME Problems/Problem 24"

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==Problem==
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In triangle <math>ABC</math>, <math>\angle C = \theta</math> and <math>\angle B = 2\theta</math>, where <math>0^{\circ} < \theta < 60^{\circ}</math>. The circle with center <math>A</math> and radius <math>AB</math> intersects <math>AC</math> at <math>D</math> and intersects <math>BC</math>, extended if necessary, at <math>B</math> and at <math>E</math> (<math>E</math> may coincide with <math>B</math>). Then <math>EC = AD</math>
  
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<math>
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\textbf{(A)}\ \text{for no values of}\ \theta \qquad
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\textbf{(B)}\ \text{only if}\ \theta = 45^{\circ} \\
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\textbf{(C)}\ \text{only if}\ 0^{\circ} < \theta \leq 45^{\circ} \qquad
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\textbf{(D)}\ \text{only if}\ 45^{\circ} \leq \theta \leq 60^{\circ} \\
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\textbf{(E)}\ \text{for all}\ \theta\ \text{such that}\ 0^{\circ} < \theta < 60^{\circ}
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</math>
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==Solution==
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nothing yet :(
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==See Also==
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{{AHSME box|year=1975|num-b=23|num-a=25}}
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{{MAA Notice}}

Latest revision as of 16:54, 19 January 2021

Problem

In triangle $ABC$, $\angle C = \theta$ and $\angle B = 2\theta$, where $0^{\circ} < \theta < 60^{\circ}$. The circle with center $A$ and radius $AB$ intersects $AC$ at $D$ and intersects $BC$, extended if necessary, at $B$ and at $E$ ($E$ may coincide with $B$). Then $EC = AD$

$\textbf{(A)}\ \text{for no values of}\ \theta \qquad \textbf{(B)}\ \text{only if}\ \theta = 45^{\circ} \\ \textbf{(C)}\ \text{only if}\ 0^{\circ} < \theta \leq 45^{\circ} \qquad \textbf{(D)}\ \text{only if}\ 45^{\circ} \leq \theta \leq 60^{\circ} \\ \textbf{(E)}\ \text{for all}\ \theta\ \text{such that}\ 0^{\circ} < \theta < 60^{\circ}$

Solution

nothing yet :(

See Also

1975 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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